In an entire year of probability theory coursework at the graduate level, there was only one time when functional analysis seriously appeared. That was ergodic theory. Now that my self-studies have carried me away to Feller processes, it has shown up again, and some serious analysis as opposed to combinatorics and elementary measure theory has begun to show up. Are there any other points where functional analysis or operator theory plays a big role?
More specifically, I have noticed that the notion of a contraction semigroup, which is important in Feller processes, can be generalized for contractions on a banach space. (In the Feller theory, the correspondence of generator, semigroup, and Feller process occurs where the semigroup is defined on the vanishing-at-infinity continuous functions on some locally compact separable metric space.) Is this merely for curiosity's sake, or is there some use of this in probability theory or elsewhere? (Don't contraction semigroups have something to do with the Feynman path integral characterization of quantum mechanics?)
Best Answer
To your last point: see the wondeful paper by Nelson on the Feynman path integral. He uses semigroup techniques to make the formalism rigorous.
As a side, unitary groups play a fundamental role in quantum mechanics because of the canonical commutation relations.
For more applications of semigroup theory apart from probability, see the outline in Chapter VI of Engel and Nagel.